Dreadnought
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Okay, so for example would g(-1) be 2?
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The Math Help Thread
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Okay, so for example would g(-1) be 2?
theinternetman
Member
What does your teacher mean by his hint that the point (-1, 2) is not on the graph but (-1,4) is?
Dreadnought
Member
I honestly don't know
Your teacher is trying to teach you about the concept of discontinuities. In this case he is showing you a jump discontinuity which is actually a very important concept. A non filled in point indicated a discontinuity. In this case g(x) is the following piece wise function.
theinternetman
Member
Great! That just means we have to step back a little bit. It would be impossible to do this problem without understanding how a graph is defined. A graph of a function g(x) is the set of points (x, g(x)) plotted on two axes. So the colored curve and isolated points you see form the set of all points (x, g(x)).
Your teacher saying (-1, 4) is on the graph essentially means when x = -1, g(x) = 4.
The idea here is to see the relationship between continuity and limits, which is very important in calculus.
I see that someone has written out the piecewise equations for your g(x), but it is crucial that you can answer this question using only the information from the graph, if only for the possibility of a question like this showing up on an exam
.I've got a question regarding geometric sequences
I have a sequence of an numbers (q>0). The sum of the last n-3 numbers is 8 times the sum of the first n-3 numbers
S(n-3) = S(n-3)*8
I know already that a1 for the last n-3 numbers (right side of the equation) is a4 of the greater an sequence. My question is why?
I have a sequence of an numbers (q>0). The sum of the last n-3 numbers is 8 times the sum of the first n-3 numbers
S(n-3) = S(n-3)*8
I know already that a1 for the last n-3 numbers (right side of the equation) is a4 of the greater an sequence. My question is why?
q=2
This is the question (It's originally written in Hebrew so perhaps my translation was bad) solved by someone else
I honestly didn't understand a thing
sryNo *I*'m sorry... It's always hard because you don't know how much people know about the subject.I honestly didn't understand a thing
When you have a geometric sequence, you go from a number to the following one using a multiplication by a common term, right?
Like
2 > 6 > 18 > 54 > 162 > ... (x 3)
3 > -6 > 12 > -24 > 48 > ... (x -2)
256 > 64 > 16 > 4 > ... (x 1/4)
Let's name the term q... (q=3, -2 and 0.25 in the above examples)
So when you want to go from a number to the following one, you multiply by q: 54 is 18x3, -24 is 12x-2, etc.
Now imagine that you want to jump directly 3 numbers: you'll multiply by q x q x q
For example, in the first sequence, you go from 2 to 54 and 6 to 162 by multiplying them by 3x3x3 = 27
Still following me this time?
Now, your problem:
You compare the sum of the n-3 first terms, so
the 1st + the 2nd + the 3rd + ... + the (n-3)th
To the sum of the n-3 last terms, so
the 4th + the 5th + the 6th + ... + the
thThose sums have the same number of terms (n-3)
To change 1th into 4th, or the 2nd into 5th, or (n-3)th into
th, we need to multiply by q x q x q, OK?So
the 4th + the 5th + the 6th + ... + the
th= the 1st x q x q x q + the 2nd x q x q x q + the 3rd x q x q x q + ... + the (n-3)th x q x q x q
= (the 1st + the 2nd + the 3rd + ... + the (n-3)th) x q x q x q
So q x q x q = 8, according to your problem.
That means q=2
Of course, there's shorter ways to solve it (I detailed so that you can understand how it works, hopefully).
One of it is knowing the formula that gives you the sum of p terms of a geometric sequence of reason q, whose first term is uo :
S = uo x (1-q^n) / (1-q)
So the first sum is = 1th term in the sequence x (1-q^(n-3)) / (1-q)
The second sum is = 4th term in the sequence x (1-q^(n-3)) / (1-q)
And 4th term = 1st term x q x q x q
So again, you get q x q x q = 8, thus q = 2.
More understandable?
Edit: corrected a type in parens
Thank you for putting so much effort into explaining this.
I understood nearly everything.
The issue I'm still struggling with is understanding why the sum of n-3 first terms ends at the 3rd and the sum of n-3 last terms start with the 4th
Well... you can put a number on all values in the sequence, like this:
1 2 3 4 5 ... (n-4) (n-3) (n-2) (n-1) n
There's n values in the sequence.
If you take the (n-3) first elements in the sequence, that's all of them barring 3, the 3 last ones, (n-2)th (n-1)th and
th. So it's 1 2 3 4 5 ... (n-4) (n-3)It doesn't end at the 3rd but at the (n-3)th.
Now, if you take the (n-3) last elements, the 3 missing ones are the first three (the 1st, 2nd and 3rd), so it's 4 5 ... (n-5) (n-4) (n-2) (n-1) n
So the last (n-3) values in the sequence of n values begin with the 4th one.
Well... you can put a number on all values in the sequence, like this:
1 2 3 4 5 ... (n-4) (n-3) (n-2) (n-1) n
There's n values in the sequence.
If you take the (n-3) first elements in the sequence, that's all of them barring 3, the 3 last ones, (n-2)th (n-1)th and
It doesn't end at the 3rd but at the (n-3)th.
Now, if you take the (n-3) last elements, the 3 missing ones are the first three (the 1st, 2nd and 3rd), so it's 4 5 ... (n-5) (n-4) (n-2) (n-1) n
So the last (n-3) values in the sequence of n values begin with the 4th one.
Thank you.
∈ means something on the left hand side is in the set of things defined by the right hand side. (0,1) is an interval. Notation can vary, but generally "(" and ")" mean the numbers aren't included in the interval. So it's saying a and b are both between 0 and 1. If non-integers don't really make sense in the problems context then it probably just means a and b are either 0 or 1.
The context of the problem is that its an optimization problem. Max (x,y) = x^a + ky^b, with k>0 and a,b ∈ (0,1)
To be honest, I'm not sure where to start.
Edit: Well that's not entirely true, but the way the problem is written is weird to me.
Interval makes sense. If it was a set, it would be {0, 1}
AntraxSuicide
Banned
The idea is that you can only have exponents a and b between 0 and 1 (that is, they're roots and not powers, in a simple interpretation).
The One Who Knocks
Member
Has anybody any good references for information on Z-matrices (matrices which are positive on the diagonal, negative off the diagonal)? I'm currently working on a problem and I've managed to break it down into simply showing the matrix I'm working on is a Z-matrix to prove a decently-sized result, but I'm not very familiar with these matrices.
I have a feeling it's a very problem-dependent topic as I've found a paper which details some properties of these matrices, but not much in terms of techniques that can be helpful to proving some matrix is in fact a Z-matrix. If anybody knows of any interesting papers that revolve around showing something is a Z-matrix, or any books which detail them, it would be very much appreciated.
I have a feeling it's a very problem-dependent topic as I've found a paper which details some properties of these matrices, but not much in terms of techniques that can be helpful to proving some matrix is in fact a Z-matrix. If anybody knows of any interesting papers that revolve around showing something is a Z-matrix, or any books which detail them, it would be very much appreciated.
a1, a2, a3, a4...
an > 0
-1<q<1
a1^2+a2^2+a3^2+a4^2... = A
(a1+a2+a3+a4...)^2 = B
I need to find q and a1 for the original sequence
answers are q = B-A/B+A
a1 = (2A square root B)/A+B
I can work out that the original sequence and B share a1 and q, and for A it is a1^2 and q^2
Other than that, I can't seem to connect the dots and reach a solution
an > 0
-1<q<1
a1^2+a2^2+a3^2+a4^2... = A
(a1+a2+a3+a4...)^2 = B
I need to find q and a1 for the original sequence
answers are q = B-A/B+A
a1 = (2A square root B)/A+B
I can work out that the original sequence and B share a1 and q, and for A it is a1^2 and q^2
Other than that, I can't seem to connect the dots and reach a solution
cpp_is_king
Member
Pretty sure it's a (infinite) geometric sequence... Especially with the -1<q<1 hint that simplify the sum of the terms.
B = (a1 / (1-q))² since it's the square of the sum of terms of a geometric sequence starting with a1
A = a1² / (1-q²) since it's the sum of terms of a geometric sequence of "multiplier" q² whose first term is a1²
Which means B/A = (1-q²)/(1-q)²
q = (B-A)/(B+A) is indeed a solution of this second order equation, so I'm sure I'm correct.
Yes, it's an infinite geometric sequence forget to mention that.
Koren, I managed to write down the sum of terms for A and B but I'm why is dividing them one way or another is the expression of q for the sequence?
Also, shouldn't it be (1-q)²/(1-q²) = (B-A)/(B+A)?
I'm sorry, but I don't understand where you're going...
Why?
I got B/A = (1-q²)/(1-q)² by dividing B = (a1 / (1-q))² by A = a1² / (1-q²)
I did that because you don't know what a1 is, so it's an easy way to get rid of it.
Or you can write
B = (a1 / (1-q))² => a1² = B (1-q)²
and
A = a1² / (1-q²) => a1² = A (1-q²)
This means B (1-q)² = A (1-q²)
which is the same...
This way, you have (B+A) q² - 2B q + (B-A) = 0. Which you can solve for q.
The Last Wizard
Member
This is how you get slope right?
slope = (y1 - y2) / (x1 - x2)
slope = (y1 - y2) / (x1 - x2)
Right
MiloHoffman
Member
Himynameischris
Member
Can someone help me explain this problem? I found the answer by using my notes from class but I don't really understand how to get the answer
When the teacher did a similar problem on the board he just divided the .2 by 2 and found the answer (the numbers were different in his example) but I don't really understand why he did that, is that true for any numbers I get? I've tried looking through the books examples but they aren't any clearer.
When the teacher did a similar problem on the board he just divided the .2 by 2 and found the answer (the numbers were different in his example) but I don't really understand why he did that, is that true for any numbers I get? I've tried looking through the books examples but they aren't any clearer.
Rich Uncle Skeleton
Member
No you can't just divide 0.2 by something to always get the answer. It depends very much on the function f. In this problem you need to use the graph to figure it out.Can someone help me explain this problem? I found the answer by using my notes from class but I don't really understand how to get the answer
When the teacher did a similar problem on the board he just divided the .2 by 2 and found the answer (the numbers were different in his example) but I don't really understand why he did that, is that true for any numbers I get? I've tried looking through the books examples but they aren't any clearer.
The idea is this: If x-values very close to 1 are plugged into f you get outputs f(x) very close to f(1) = 1 (since f is continuous). What you need to do here is see how far those x-values can stray from 1 while keeping the y-values (outputs) close enough to f(1) = 1. If you move x to the left from 1 you can go 0.3 units before outputs climb to 1.2 (which is distance .2 from 1) -- this is the significance of the red line on the left. If you move x to the right you can go 0.1 units before outputs fall to 0.8 (which again is .2 from 1), and this is the significance of the red line on the right. So if x is less than 0.1 from 1 then regardless of if it's to the right or to the left, you're guaranteed to have 0.8 < f(x) < 1.2, which is what you want. Hope this helps.
Himynameischris
Member
So for this particular questions I just look at .7 and 1.1 and use the one nearest 1?
That's basically it...
Think that all x values between 0.7 and 1.1 give an acceptable result. Now, imagine a balloon whose center is on 1.0, and slowly inflate it.
0.98-1.02 fits
0.95-1.05 fits
0.9-1.1 fits
0.85-1.15 don't fit anymore in [0.7, 1.1]
So yes, the idea is looking for the closest edge from 1.0
Allonym
There should be more tampons in gaming
Can someone help me with this problem? It concerns limits and basically is this;
limit
x-->0 (1-cos √x)/x
so I came up with (sin√x)/x * (sin√x)/(1+cos√x) and I'm not certain what to do from this point on. From here do I substitute the zero in? I feel like if I did that I'd get an indeterminate or undefined value, right? Any help would be appreciated.
limit
x-->0 (1-cos √x)/x
so I came up with (sin√x)/x * (sin√x)/(1+cos√x) and I'm not certain what to do from this point on. From here do I substitute the zero in? I feel like if I did that I'd get an indeterminate or undefined value, right? Any help would be appreciated.
You're almost there. Make the substitution u = sqrt(x) and use the fact that lim_{u -> 0} sin(u) / u = 1.
Then,
lim_{x -> 0+} sin^{2}(sqrt(x)) / (x * (1 + cos(sqrt(x)))
= lim_{u -> 0+} sin^{2}(u) / (u^2 * (1 + cos(u)))
= [lim_{u -> 0+} sin(u) / u]^2 * lim_{u -> 0+} 1 / (1 + cos(u))
= (1)^2 * 1/2
= 1/2.
You can also use L'Hopital's Rule to arrive at the answer.
Himynameischris
Member
How would I finish this problem?
jellies_two
Member
How would I finish this problem?
Isn't that the "location theorem" ?
"Let f(x) be a polynomial, all of whose coefficients are real numbers. If a and b are real numbers such that f(a) and f(b) have opposite signs, then the equation f(x) = 0 has at least one real root between a & b."
Himynameischris
Member
It's supposed to be the intermediate value theorem but I'm kinda stuck on how to prove there is a root.
Biggest-Geek-Ever
Member
You won't be able to figure exactly what the root, so the best you can do is prove the existence of one.The intermediate value theorem says:
Suppose that f(x) is continuous on [a, b] and let M be any number between f(a) and f(b). Then there exists a number c such that
(i) a < c < b, and
(ii) f(c) = M
Your f(x) is a polynomial, so it is continuous. Since 0 is between f(0) = -1 and f(1) = 9, the IVT tells us there is some value c such that 0 < c < 1 and f(c) = 0. This value c is hence a root.
Himynameischris
Member
Hmm that's what I put on my test but the teacher marked it wrong anyways, I have to ask him what I did wrong then.
Thanks!
Himynameischris
Member
Am I missing something here? I've gone through the problem several times but I can't seem to get the answer :/
I've inputted all three variations of the answer but I get it wrong...
I've inputted all three variations of the answer but I get it wrong...
Your answer is %100 correct.Am I missing something here? I've gone through the problem several times but I can't seem to get the answer :/
I've inputted all three variations of the answer but I get it wrong...
Edit: ...If it weren't for changing a value (y1) that I missed
Except that y1 is not 2 (or I'm really, really dumb), as I was suggesting above... most probably the reason the result is rejected.
But I'm sure it's just a typo, the rest is indeed sound and correct.
Himynameischris
Member
It is 2 sq rt 3/3, but that reduces to 2, should I just keep it as 2 sq rt 3/3?
It's not, as you suggested above, my bad. Didn't think that op might have changed a value that's already been given (y1) somewhere down the line.
Incorrect, you can calculate y1 even if it hadn't been given, y1 = sec(pi/6) = 1/cos(pi/6) = 1/[sqrt(3)/2] = (2/3) * sqrt(3)
Himynameischris
Member
Might be a glitch then, I'll have to ask my teacher.
No, it's 2 sqrt(3) / 3, which reduces to 2 / sqrt(3) and not 2.
But I understand, you probably read 2 sqrt(3/3), that explains the issue.
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